Question

The number of distinct real value of $\lambda$ for which the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{{\lambda }^2}$ and $\frac{x-3}{1}=\frac{y-2}{{\lambda }^2}=\frac{z-1}{2}$ are coplanar is:

• A. $4$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is D $3$

From the question we find the given matrix

$\left|\begin{array}{ccc}3-1& 2-2& 1+3\\ 1& 2& {\lambda }^2\\ 1& {\lambda }^2& 2\end{array}\right|=0$

$2\left(4-{\lambda }^4\right)+4\left({\lambda }^2-2\right)=0$

$4-{\lambda }^4+2\left({\lambda }^2-2\right)=0$

$-{\lambda }^4+2{\lambda }^2=0$

${\lambda }^2\left(2-{\lambda }^2\right)=0$

${\lambda }^2\left(2-{\lambda }^2\right)=0$

$\lambda =\pm \sqrt{2}$

$\lambda =0$

Hence, the option $\left(D\right)$ is the correct answer.